4. Maximizing the unanticipated rate of return among immunized bond portfolios

The natural question arises of how to select the "best" portfolios among those which are (have been) protected (immunized) against admissible shifts (movements) of interest rates? In finance, by best portfolios are meant those which yield the highest rate of return (the highest increase in the present value of a BP), resulting from a sudden shift of interest rates. Below, we present the results obtained in [19]. Rewriting a sufficient and necessary condition (13) and (14) for immunization of portfolio BP generating payouts (9), one obtains:

$$q = \sum\_{i=1}^{i=m} t\_i w\_i v\_i \tag{24}$$

and

$$w\_i = \frac{a(t\_i)}{a(q)}\text{ provided } a(q) \neq 0. \tag{25}$$

For each vector <sup>v</sup> <sup>¼</sup> ð Þ <sup>v</sup>1; <sup>v</sup>2;…; vm <sup>∈</sup> Rm, the class Kv of such continuous shifts a(t) for which (25) holds was defined in [19]. When vector <sup>v</sup> <sup>¼</sup> ð Þ <sup>1</sup>; <sup>1</sup>;…; <sup>1</sup> <sup>∈</sup> Rm is used, then the corresponding class Kv comprises all parallel shifts for which a(t) = constant. We shall call Dv <sup>¼</sup> <sup>i</sup> P¼m i¼1 tiwivi the dedicated (for class Kv) duration. For zero-coupon bearing bond Bk, maturing at time tk, the dedicated duration Dvð Þ¼ Bk tk � vk since all weights wi, except for wk, are equal to 0.

Theorem 5. The immunization of a bond portfolio BP against shifts a(t) from class Kv is secured if and only if q ¼ Dvð Þ¼ BP i P¼m i¼1 tiwivi:

With s(t) standing for the current interest rates, PV s½ �¼ ð Þ� <sup>k</sup> P¼m k¼1 ck exp �s tð Þ<sup>k</sup> tk ½ � is the present value of BP. Suppose that immediately after purchasing BP, interest rates s(t) will shift to new levels s\*(t) = s(t) + a(t). Then

$$PV[s(\cdot) + a(\cdot)] = \sum\_{k=1}^{k=m} c\_k \exp\left[-s(t\_k) - a(t\_k)\right] t\_k.$$

#### 4.1. Convexity of a bond portfolio

SkðÞ¼ t

t � tkþ<sup>1</sup> tk � tkþ<sup>1</sup>

SkðÞ¼ t

106 Immunization - Vaccine Adjuvant Delivery System and Strategies

It is a nice exercise to prove the following result.

The following result is well known.

(t)|| = 0.

portfolios

and

t � tk�<sup>1</sup> tk � tk�<sup>1</sup>

Remark 5. Each continuous function a(t) defined on ½ � t0; T attains the same values as the function b tðÞ¼ að Þ� 0 S0ð Þþ t a tð Þ� <sup>1</sup> S1ð Þþ t a tð Þ� <sup>2</sup> S2ð Þþ t … þ a tð Þ� <sup>m</sup> Smð Þt (built up with (m + 1) triangular functions) at all points tk, 0 ≤ k ≤ m. Therefore, a(t) may be identified in H\* with the piecewise linear function b(t) because the distance between a(t) and b(t) in H\* is zero: ||b(t) – a

Remark 6. The Lagrange functions S0ð Þt , S1ð Þt , S2ð Þt , …, Smð Þt given by (20)–(23) constitute a

Theorem 4. The set IMMU of all shifts (continuous functions) against which a bond portfolio BP with payouts represented by (9) and the new term structure given by (12) is immunized constitutes an m-dimensional linear subspace in the (m + 1)-dimensional Hilbert space H\*.

Two examples illustrating how to identify IMMU are worked out in detail in [18], pp. 531–537. A special attention is given to continuity properties of subspace IMMU; see [18], pp. 534–537.

4. Maximizing the unanticipated rate of return among immunized bond

The natural question arises of how to select the "best" portfolios among those which are (have been) protected (immunized) against admissible shifts (movements) of interest rates? In finance, by best portfolios are meant those which yield the highest rate of return (the highest increase in the present value of a BP), resulting from a sudden shift of interest rates. Below, we present the results obtained in [19]. Rewriting a sufficient and necessary condition (13) and (14)

> <sup>q</sup> <sup>¼</sup> <sup>X</sup> i¼m

> > i¼1

tiwivi (24)

a qð Þ provided a qð Þ 6¼ <sup>0</sup>: (25)

for immunization of portfolio BP generating payouts (9), one obtains:

vi <sup>¼</sup> a tð Þ<sup>i</sup>

base for Hilbert space H\* of all admissible (continuous) shifts defined on ½ � t0; T .

, t∈ tk�<sup>1</sup>; tk ½ �, 1 ≤ k ≤ m � 1, (22)

, t∈ tk ½ � ; tkþ<sup>1</sup> ; SkðÞ¼ t 0 elsewhere in ½ � t0; T : (23)

Set Cvð Þ¼ BP <sup>1</sup> 2 Pm k¼1 tk <sup>2</sup>wkvk <sup>2</sup> and call it dedicated (for class Kv) convexity of portfolio BP; for more details, see [19], p. 105. It is easy to notice that convexity of a zero-coupon bearing bond maturing at tk is given by the formula Cv <sup>¼</sup> <sup>1</sup> 2 tk 2vk 2. It was proved in [19], p. 106, that so-called unanticipating rate of return resulting from a shift a(t) of interest rates sð Þ� is given by the formula:

$$\frac{PV[\mathbf{s}(\cdot) + a(\cdot)] - PV[\mathbf{s}(\cdot)]}{PV[\mathbf{s}(\cdot)]} = -D\_v(BP)a(\mathbf{q}) + \mathbf{C}\_v(BP)a^2(\mathbf{q}) + \sum\_{k=1}^{k=m} \mathbf{O}[a(\mathbf{t}\_k)]a(\mathbf{t}\_k)^2 \tag{26}$$

where lim O(a) = 0 when a ! 0. Taking into account that a tð Þ<sup>k</sup> are small numbers of order 0.1% = 0.001, one concludes that the third term in (26) is really very small. Since each immunized bond portfolio BP satisfies Dvð Þ¼ BP q, the maximal unanticipating rate of return among immunized portfolios will be achieved when dedicated convexity Cvð Þ BP will be as high as possible.

Assumption 1. All zero-coupon bearing bonds Bk, which form a bond portfolio BP and mature at tk, have mutually different dedicated durations, that is, Dv Bj � � 6¼ Dvð Þ Bn if and only if j 6¼ n, that is, tj � vj 6¼ tn � vn \$ j 6¼ n.

Definition 4. Following [20], p. 552, a bond portfolio BP is said to be a barbell strategy (barbell portfolio) if it is built up of two bonds, say B<sup>1</sup> , B<sup>2</sup> with significantly different dedicated durations Dv <sup>1</sup> and Dv 2 . On the other hand, BP is said to be a focused strategy (focused portfolio) if it consists of several bonds whose dedicated durations Dv <sup>j</sup> are centered around duration of the liability (q in our context).

Theorem 6. (see [19], Theorem 1). If Assumption 1 holds then the bond portfolio BP\* with the highest unanticipated rate of return is a barbell strategy built up of zero-coupon bearing bonds Bs , Bl with minimal and maximal dedicated durations. The weights xs and xl, expressing the amounts of payments resulting from B<sup>s</sup> and Bl , are given by formulas:

$$\overline{\mathbf{x}}\_{s} = \frac{t\upsilon\eta - \eta}{t\upsilon\eta - t\_{s}\upsilon\_{s}},\\\overline{\mathbf{x}}\_{l} = \frac{\eta - t\_{s}\upsilon\_{s}}{t\_{l}\upsilon\_{l} - t\_{s}\upsilon\_{s}},\\\overline{\mathbf{x}}\_{k} = 0 \text{ for } k \neq s, k \neq l. \tag{27}$$

in medicine of the duration concept defined for the first time by Macaulay (in 1938) and

Practical Finance Strategies in Immunization http://dx.doi.org/10.5772/intechopen.78719 109

Let us formulate the following hypothesis: the higher values (levels) of Z, the more healthy is a

In the financial immunization context, there is a fixed date q when BP must attain at least a certain value L, called liability. In the medical context, one might say that there is a fixed date q

In the financial theory context, when interest rates s(t) change due to a shift a(t), that is, s tðÞ! s tð Þþ a tð Þ, then the FV of BP at date q may fall below L dollars. In the medical context,

We still do not know what should (could) be substituted for interest rates s(t), knowing that

Using the concept of duration (and dedicated duration), we identified the set IMMU of all shifts (diseases) a(t) against which BP is immunized. By means of notion of duration and convexity (dedicated convexity), we determined the best immunizing portfolios for a large class of shifts (continuous functions). In the financial context, the best portfolios meant portfolios generating the highest (unanticipated) rate of return. In the medical context, the best would probably

[1] Macaulay F. Some theoretical problems suggested by the movements of interest rates, bond yields and stock prices in the U.S. since 1856. Working paper. National Bureau of

[2] Redington F. Review of the principle of life office valuations. Journal of the Institute of

[3] Fisher L, Weil R. Coping with the risk of interest rate calculations: Returns for bondholders from naive and optimal strategies. Journal of Business. 1971;44:408-431. DOI:

[4] Bierwag G. Immunization, duration and term structure of interest rates. Journal of Finan-

cial and Quantitative Analysis. 1977;12:725-742. DOI: 10.2307/2330253

independently by Redington (in 1952); see Formula (1).

when the quality of human health must attain at least a certain level L.

the appearance of disease may cause a deterioration of health at date q.

changes (movements, shifts) in interest rates mean a disease.

mean the fastest rate of health improvement.

Address all correspondence to: l.zaremba@vistula.edu.pl

Economic Research; New York; 1938

Actuaries. 1952;18:286-340

10.1086/295 402

Academy of Finance and Business Vistula, Warsaw, Poland

Author details

Leszek Zaremba

References

human body (a human body organ).

Comment 1. Suppose that instead of dedicated duration and dedicated convexity, we employ the classic notions of duration and convexity derived for additive shifts only. Then, vl � 1 and consequently Eq. (27) reduces to simpler, say classic, formulas xs <sup>¼</sup> tl�<sup>q</sup> tl�ts , xl <sup>¼</sup> <sup>q</sup>�ts tl�ts . The natural question arises of how much the weights given by Eq. (27) differ from the classic ones.

Finally, another interesting question arises, to what extend does the dedicated duration of the best immunized portfolio BP\* differ from its Macaulay's counterpart? That is, what is the difference between Dv BP<sup>∗</sup> ð Þ¼ tswsvs <sup>þ</sup> tlwlvl and D BP<sup>∗</sup> ð Þ¼ tsws <sup>þ</sup> tlwl with vs <sup>¼</sup> a tð Þ<sup>s</sup> a qð Þ, vl <sup>¼</sup> a tð Þ<sup>l</sup> a qð Þ? It is easy to observe that when a shift a(t) affects the current interest rates in a similar manner at all or many points t1, t2, t3, …, tm, then there is a good chance that vs ≈ 1 ≈ vl, and consequently, the difference between the dedicated duration Dv and the classic one will be very small.

For a specific situation, when shifts a(t) of interest rates s(t) satisfied the "proportionality" condition a tð Þ<sup>k</sup> <sup>1</sup>þs tð Þ<sup>k</sup> <sup>¼</sup> constant (for details, see [21]), the maximal convexity and formula for the best immunizing bond portfolio was determined by means of Kuhn-Tucker conditions (pp. 139–140 in [21]). A formula for the resulting unanticipating rate of return was derived (pp. 141–142) and illustrating with an example (p. 143).
